Number of Spanning Trees of Circulant Graphs C 6n and their Applications

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1 Joural of Mathematics ad Statistics 8 (): 4-3, 0 ISSN Sciece Publicatios Number of Spaig Trees of Circulat Graphs C ad their Applicatios Daoud, S.N. Departmet of Mathematics, Faculty of Sciece, El-Miufiya iversity, Shebee El-Kom, Egypt Departmet of Applied Mathematics, Faculty of Applied Sciece, Taibah iversity, Al-Madiah, K.S.A. Abstract: Problem statemet: The umber of spaig trees of a graph G is usually deoted by τ(g). A circulat graph with vertices ad jumps is C (a,..,a ). Approach: I this study the umber τ(g) of spaig trees of the circulat graphs C with some o-fixed jumps such as C (, ), C (,, ), C (,, 3), C (, 3), C (,,, 3), are evaluated usig Chebyshev polyomials. A large umber of theorems of umber of the spaig trees of circulate graphs C are obtaied. Results: The umber τ(g) of spaig trees of the circulat graphs C (, ), C (,, ), C (,, 3), C (,, 3), C (,,, 3), C (,, 3), C (, 3, ), C (, 3, 4), C (,, 3, 4), C (,, 3, ), C (, 3, 4, ) ad C (,, 3, 4, ) are evaluated. Coclusio: The umber of spaig trees τ(g) i graphs (etwors) is a importat ivariat. The evaluatio of this umber ad aalyzig its behavior is ot oly iterestig from a mathematical (computatioal) perspective, but also, it is a importat measure of reliability of a etwor ad desigig electrical circuits. Some computatioally hard problems such as the travellig salesma problem ca be solved approximately by usig spaig trees. Due to the high depedece of the etwor desig ad reliability o the graph theory we itroduced the followig importat theorems ad their proofs. Key words: Spaig trees, circulat graphs, chebyshev polyomials, irchhoff matrix INTRODCTION The umber of spaig trees τ(g) i graphs (etwors) is a importat ivariat. The evaluatio of this umber ad aalyzig its behavior is ot oly iterestig from a mathematical (computatioal) perspective, but also, it is a importat measure of reliability of a etwor ad desigig electrical circuits. Some computatioally hard problems such as the travellig salesma problem ca be solved approximately by usig spaig trees. I this study we cosider fiite udirected graph with o loops or multiple edges. Let G be such a graph of vertices. A spaig tree for a graph G is a sub graph of G that is a tree ad cotais all vertices of G.The umber of spaig trees of G, deoted by τ(g), is the total umber of distict spaig sub graphs of G that are trees. A classic result of Kirchhoff (874) a be used to determie the umber of spaig trees for G (V, E). Let V (v,v,,v ). The Kirchhoff matrix H is defied as characteristic matrix H D-A, where, D is the diagoal matrix of the degrees of G ad A is the adjacecy matrix of G, H [a ij ] is defied as follows: (i) a ij -, whe v i ad v j are adjacet ad i j, (ii) a ij equal the degree of vertex v i if i j ad (iii) a ij 0 otherwise. All of co-factors of H are equal to τ(g). There are more tha method for calculatig τ(g). Let µ µ. µ p deote the eigvalues of H matrix of a p poit graph. It ca be easily show that µp 0. Kelmas ad Cheloov (974) proved that Eq. : τ (G) µ p p () The formula for the umber of spaig trees i a d-regular graph G ca be expressed as p τ (G) (d µ ) where µ 0 d, µ, µ,.,µ p- are the p eigevalues of the correspodig adjacecy matrix of the graph. The circulat graphs are a importat class of graphs, which are used i the desig of local area etwors. 4

2 J. Math. & Stat., 8 (): 4-3, 0 Let a a... a, where ad a i (i,,) are positive itegers. A udirected circulat graph C (a,a,,a ) is a regular graph whose set of vertices is V {0,,,,-} ad whose set of edges is E {(i, i+a j (mod ))/i 0,,,,-, j,,,}. If a, the C (a,a,,a ) is a -regular graph; if a, the it is a (-)-regular oe, Niolopoulos ad Papadopoulos (004). The simplest circulat graph is the vertex cycle C (). The well ow formula τ (, )) F, where F is the th Fiboacci umber defied by the recursive relatio F F - +F -,, 3.with iitial coditio F 0 0, F, Kleima ad Golde (975). The formulas for τ (, 3)), τ (,4)) ad more geeral results have recetly bee obtaied by Yog ad Ataja (997). The formulas of τ (, )), τ 3 (,)), τ 4 (,)) ca be foud i Zhaga et al. (005). It ca the be show from this recursio that by expadig det A (x) oe gets Eq. 5: (x) det(a (x)), (5) Furthermore by usig stadard methods for solvig the recursio (4), oe obtais the explicit formula Eq. 6: (x) [(x + x ) x + + (x x ) ], (6) where, the idetity is true for all complex x (except at x ±where the fuctio ca be tae as the limit). The defiitio of (x) easily yields its zeros ad it ca therefore be verified that Eq. 7: j (x) (x cos ) (7) Chebyshev polyomial: Now we itroduce some some relatios cocerig Chebyshev polyomials of the first ad secod id which we use it i our computatios. We begi from their defiitios, Zhaga et al. (000). Let A (x) be matrix such that: Oe further otes that Eq. 8: ( x) ( ) (x) (8) These two results yield aother formula for (x): x 0 x 0 A (x) 0 0 x where all other elemets are zeros. Further we recall that the Chebyshev polyomials of the first id are defied by Eq. : T (x) cos( arccos x) () j (x) 4 (x cos ) (9) Fially, simple maipulatio of the above formula yields the followig, which also will be extremely useful to us latter Eq. 9-: x + j ( ) (x cos ) (0) 4 Furthermore oe ca show that: The Chebyshev polyomials of the secod id are defied by Eq. 3: d si( arccos x) (x) T (x) dx si(arccos x) (3) (x) [ T ] ( x ) ( x ) [ T (x )] () It is easily verified that Eq. 4: (x) x (x) + (x) 0 (4) 5 Ad: + + () T (x) [(x x ) (x x ) ]

3 J. Math. & Stat., 8 (): 4-3, 0 Lemma : The Kirchhoff matrix of the circulat graph C (a,a,,a ) has eigevalues. They are 0 ad j, j - the values: ε... ε ε... ε i s j s j s j s j where, ε e (Biggs, 993). Pluggig this ito() yields: Corollary : Theorem 3: Proof: Let ε (s,s,...,s )) s j s j s j s j ( ε... ε ε... ε ) jsi ( cos ) i 5 (,)) [( ) τ ( ) ] [( + ) + ( ) ] [( + ) + ( ) ] [( + ) + ( ) + ] i/ e. By Lemma. we have: τ ε ε ε ε j j j j (,)) [4 ] j j j j j j, / j,3 / j, j,3 j, / j,3 j [4 cos cos ] [4 cos cos ] 3 [3 cos ] [ cos ] [6 cos ] [5 cos ], j,3 / j (3 cos )( cos ) [3 cos ] j j j (6 cos )(5 cos ) (5 cos ) (6 cos ) (3 cos ) (3 cos ) Put j 6 j i the secod term, j j i the third term ad j 3 j i the fourth term for some iteger j, we get: (,)) [3 cos ] Put j 3 j i the secod term for some iteger j, we get: 6 (3 cos )( cos ) (6 cos )(5 cos ) 3 (5 cos ) (6 cos ) 3 (3 cos ) (3 cos ) ( ) ( ) () 3 ( ) ( ) ( ) ( ) 3 ( ) ( ) 5 5 [( + ) + ( ) ] [( + ) + ( ) ] [( + ) + ( ) ] [( + ) + ( ) + ] where, are used to derive the last two steps. Theorem 4: Proof: Let ε (,, ) ) [( + ) + ( ) ] [( + ) + ( ) + ] i/ e. By Lemma. we have: τ ε ε ε ε ε ε j j j j j j (,,)) [6 ] j j 4 j [6 cos cos cos ] j j j [6 cos cos cos ] 3 3 j j j [7 cos cos ] [4 cos 3,3 / j, 3 j j 3 [7 cos cos ] 3 j cos ] 3 j 4 cos cos j 7 cos cos 3

4 6 j 6 (,, ) ) [7 cos cos ] 3 4 cos cos j 6 [6 cos ], j,3 j 7 cos cos j / / 6 6 [5 cos ] [9 cos ], j,3 j, / j,3 j 6, j,3 / j [8 cos ] 6, / j, j, / j, j 6 [6 cos ] [ cos ] [9 cos ] [5 cos ] [6 cos ] 6 (6 cos )(5 cos ) 6 (8 cos ) (9 cos )(8 cos ) (6 cos ) cos [6 cos ] j 6 (9 cos ) 6 cos j ] j (6 cos ) [9 cos ] 5 cos 9 cos Put j 6 j i the secod term, j j i the third term, j 3 j i the fourth term ad j j i the sixth term for some iteger j, we get: (,, )) [6 cos ] (6 cos )(5 cos ) (9 cos )(8 cos ) [6 cos ] 3 (8 cos ) (9 cos ) 3 j ] (6 cos ) (6 cos ) [9 cos ] 3 cos j 5 6 cos ( ) 3( ) [( + ) cos 3( ) ( ) 9 cos J. Math. & Stat., 8 (): 4-3, 0 + ( ) ] [( + ) + ( ) + ] Theorem 5: (,,3)) [( + ) + ( ) ] 6 [( 3 + ) + ( 3 ) ] [( + ) + ( ) + ] Proof: Let ε i/ e. By Lemma. we have: τ ε ε ε ε ε ε j j 3j j j 3j (,,3)) [6 ] 6j [6 cos cos cos ] j [6 cos cos cos j] 3, j j [8 cos cos ] 3, / j [4 cos j j j cos ] 3 [8 cos cos ] 3 j 4 cos cos 3 j 8 cos cos 3 Put j j i the secod term for some iteger j, we get: j (,,3)) [8 cos cos ] cos cos 3 3 [7 cos ] 8 cos cos, / j,3 / j 3 3 [6 cos ] [0 cos ], j,3 j, / j,3 j, j,3 / j [9 cos ] [5 cos ] [ cos ] 3 3 [9 cos ] [6 cos ] ,3 / j,3 j 3 3,3 / j,3 j [7 cos ] (7 cos )(6 cos ) j (0 cos )(9 cos )

5 J. Math. & Stat., 8 (): 4-3, 0 (9 cos ) (0 cos ) ] (7 cos ) (7 cos ) 3 cos 3 3 [5 cos ] 5 cos j [9 cos ] 3 6 cos cos 3 Put j 6 j i the secod term, j j i the third term, j 3 j i the fourth term ad j 3 j i the sixth term for some iteger j, we get: [7 cos ] (7 cos )(6 cos ) 3 (9 cos ) 3 (0 cos )(9 cos ) (7 cos ) 3 3 [5 cos ] (0 cos ) 3 ] 3 (7 cos ) [9 cos ] 3 cos j 5 cos 6 cos j 9 cos ( ) ( ) ( 3) 3 ( ) ( ) ( ) ( 3) ( ) [( + ) + ( ) ] [( 3 + ) 6 + ( 3 ) ] [( + ) + ( ) Theorem 6: + ] 7 7 (,,3)) [( + ) + ( ) ] 6 [( + ) + ( ) + ] [( + ) + ( ) ] 8 Proof: Let i/ ε e. By Lemma. we have: τ ε ε ε ε ε ε j j 3j j j 3j (,,3)) [6 ] 4j 6j [6 cos cos cos ] [6 cos cos cosj] 3 [8 cos cos ] [4 cos 3, / j, j 3 [8 cos cos ] 3 cos ] 3 4 cos cos 8 cos cos 3 Put j j i the secod term for some iteger j, we get: (,,3)) [8 cos cos ] 3 4j 3 4 cos cos 3 3 [8 cos 4j 8 cos cos cos 4cos 3 3 cos ] 3 j 0 cos 4cos 3 3 [9 cos ] [6 cos ],3 / j,3 j 3 3,3 / j,3 j 3 3,3 / j,3 j [5 cos ] [ cos ] 3 3 [9 cos ] [6 cos ] 3 3 (6 cos ) [9 cos ] (9 cos ) 3 ( cos ) 3 [5 cos ] 3 (5 cos ) 3 [9 cos ] 3 (6 cos ) 3 3 (9 cos ) Put j 6 j i the secod ad fourth terms for some iteger j, we get:

6 J. Math. & Stat., 8 (): 4-3, 0 (6 cos ) [9 cos ] j j (9 cos ) cos [9 cos ] 6 cos 3 9 cos 3 [5 cos ] 5 cos ( ) ( ) ( ) 3 ( ) 6 7 3( ) ( ) ( ) ( ) [( + ) + ( ) ] [( + ) ( ) + ] [( + ) + ( ) ] 4 4 Theorem 7: τ (,,,3)) [( ) ( ) ] [( 3 + ) + ( 3 ) ] [( + ) ( ) + ] [( 3 + ) + ( 3 ) + ] Proof: Let ε i/ e. By Lemma. we have: (,,,3) ) [8 ε ε ε j j j 3j j j j 3j ε ε ε ε ε ] [8 cos, / j 4j 6j cos cos cos ] j [8 cos cos cos cos j] 3 3 j [0 cos cos cos ] 3 3, j j [6 cos cos cos ] 3 3 j [0 cos cos cos ] 3 3 j 6 cos cos cos 3 3 j 0 cos cos cos Put j j i the secod term for some iteger j, we get: j (,,,3)) [0 cos cos cos ] cos cos 4cos j cos cos 4cos 3 3 j [9 cos cos ] 3,3 j,3 / j j [8 cos cos ] 3 (8 cos ) ( cos ) 3 3 ( cos ) (6 cos ) ,3 / j,3 j 3 3,3 / j,3 j j 8 cos cos j 3 [9 cos cos ] j 3 j 9 cos cos 3 3 cos 3 3 [8 cos ] 8 cos j [ cos ] 3 6 cos cos 3 Put j 3 j i the secod ad fourth terms for some iteger j, we get: j (,,,3)) [9 cos cos ] 3 cos 3 [8 cos ] j 8 cos cos j 8 cos j 3 3 j 9 cos cos j [ cos ] 6 cos 3 cos [ 8 cos ] [7 cos ], / j,3 / j, j,3 j [ cos ] [0 cos ], / j,3 j, j,3 / j [0 cos ] [6 cos ] [ cos ] [7 cos ], / j, j, / j, j

7 J. Math. & Stat., 8 (): 4-3, 0 cos 3 [8 cos ] j 3 8 cos [8 cos ] 3 [ cos ] 6 cos 3 cos (8 cos )(7 cos ) (0 cos ) ] ( cos )(0 cos ) (8 cos ) 6 cos [0 cos ] j ( cos ) 0 cos j j (8 cos ) [ cos ] 7 cos cos cos 3 [8 cos ] 8 3 cos 3 [ cos ] 6 cos 3 j cos Put j 6 j i the secod term, j j i the third term, j 3 j i the fourth term ad j j i the sixth term for some iteger j, we get: (,,,3)) [8 cos ] (8 cos )(7 cos ) 3 (0 cos ) 3 ] ( cos )(0 cos ) (8 cos ) 3 6 cos [0 cos ] ( cos ) j 0 cos (8 cos ) [ cos ] 7 cos cos cos 3 [8 cos ] 8 cos j 3 3 [ cos ] 6 cos 3 cos 5 7 ( ) ( ) 3( 3) ( 3) ( ) 3( ) ( 3) ( 3) 7 5 [( + ) + ( ) + ] [( + ) Theorem : ( ) + ] [( 3 + ) + ( 3 ) + ] [( 3 + ) + ( 3 ) ] Now we coclude that a few more applicatios (proofs omitted). Theorem 8: (,,3)) [( + ) + ( ) ] [( + ) + ( ) + ] [( + ) + ( ) ] [( + ) + ( ) ] 7 7 [( + ) + ( ) ] Theorem 9: (,3,)) [( + ) + ( ) ] 4 Theorem 0: [( + ) + ( ) ] (,3,4)) [( + ) + ( ) ] [( + ) + ( ) + ] 3 3 [( + ) + ( ) ] [( + ) + ( ) ] 7 7 [( + ) + ( ) ] 4 4

8 J. Math. & Stat., 8 (): 4-3, 0 (,,3,4)) [( + ) + ( ) ] 7 5 [( ) ( ) ] [( ) Theorem : 7 5 ( ) ] [( + ) + ( ) ] 7 7 (,,3,)) [( + ) + ( ) ] 7 7 [( + ) + ( ) + ] [( + ) + ( ) + ] [( + ) + ( ) ] [( + ) + ( ) + ] [( + ) + ( ) ] Theorem 3: (,3,4,)) [( + ) + ( ) + ] 7 7 [( + ) + ( ) ] [( + ) + ( ) ] [( + ) + ( ) ] [( + ) + ( ) + ] [( + ) + ( ) + ] Theorem 4: 8 8 (,,3,4,) ) [( 3 + ) + ( 3 ) ] 4 4 [( + ) + ( ) ] [( + ) + ( ) + ] [( + ) + ( ) ] [( + ) + ( ) ] [( + ) + ( ) + ] 4 4 [( + 3) + ( 3) + ] 4 4 ACKNOWLEDGMENT The researchers is deeply idebted to the team of wor at deaship of scietific research at Taibah uiversity, for their cotiuous helps ad the ecouragemet me to fialize this wor. This research wor was supported by a grat No. (65/43) from the deaship of the scietific research at Taibah uiversity, Al-Madiah Al-Muawwarah, K.S.A. REFERENCES Biggs, N., 993. Algebraic Graph Theory. d Ed., Cambridge iversity Press, Cambridge, ISBN: , pp: 05. Kelmas, A.K. ad V.M. Cheloov, 974. A certai polyomials of a graph ad graphs with a extermal umber of trees J. Comb. Theory, 6: Kirchhoff, G.G., 874. ber die auflosug der gleichuge, auf welche ma beider utersuchug der lieare verteilug galvaischer strme gefhrt wird. A. Phys. Chem., 7: Kleima, D.J. ad B. Golde, 975. Coutig trees i a certai class of graphs. Amer Math. Mothly, 8: Niolopoulos, S.D. ad C. Papadopoulos 004. the umber of spaig trees i K -Complemets of quasi-threshold graphs. Graphs Combiatorics, 0: DOI: 0.007/s x Yog, X.R. ad T. Ataja, 997. The umber of 3 spaig trees of the cubic cycle CN C ad the 4 quadrauple cycle C N. Discrete Math, 69: Zhaga, Y., X. Yogb ad M.J. Golic, 000. The umber of spaig trees i circulat graphs. Discrete Math., 3: Zhaga, Y., X. Yogb ad M.J. Golic, 005. Chebyshev polyomials ad spaig tree formulas for circulat ad related graphs. Discrete Math., 98: DOI: 0.06/j.disc